Abstract: The number of steps any classical computer requires in order to find the
prime factors of an $l$-digit integer $N$ increases exponentially with $l$, at
least using algorithms known at present. Factoring large integers is therefore
conjectured to be intractable classically, an observation underlying the
security of widely used cryptographic codes. Quantum computers, however, could
factor integers in only polynomial time, using Shor's quantum factoring
algorithm. Although important for the study of quantum computers, experimental
demonstration of this algorithm has proved elusive. Here we report an
implementation of the simplest instance of Shor's algorithm: factorization of
${N=15}$ (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a
molecule as quantum bits, which can be manipulated with room temperature liquid
state nuclear magnetic resonance techniques. This method of using nuclei to
store quantum information is in principle scalable to many quantum bit systems,
but such scalability is not implied by the present work. The significance of
our work lies in the demonstration of experimental and theoretical techniques
for precise control and modelling of complex quantum computers. In particular,
we present a simple, parameter-free but predictive model of decoherence effects
in our system.